Pick's Theorem

Date: 9/4/96 at 22:31:47
From: -={Mirage}=-
Subject: Pick's Theorem
Would you happen to know Pick's theorem? This has been bugging me for
almost a year.
Thanks,
Jeff

Date: 9/5/96 at 13:22:54
From: Doctor Tom
Subject: Re: Pick's Theorem
Hi Jeff,
Sure.
On a grid of points (like on graph paper) draw a simple polygon (no
holes, no lines crossing each other), so that all the vertices lie on
the grid. Then count the number of points completely inside (call this
number I), and count the number of points that lie on the edges or
corners of the polygon (call this number E). Then the area inside the
polygon is given by the formula: I + E/2 - 1, where the area is
measured in terms of the smallest squares you can make on the grid.
You can find a proof of this in "Introduction to Geometry," second
edition, by Coxeter (and probably in lots of other places - that's the
first advanced geometry book I hauled off my shelf).
Try a bunch of examples yourself to convince yourself of the truth.
The proof isn't too difficult either, but it's not totally elementary,
so you may have to look it up.
-Doctor Tom, The Math Forum
Check out our web site! http://mathforum.org/dr.math/